Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
To recap, the Greek concept of magnitude was somewhat tied to what a person could measure in the real world. The Pythagoreans believed that all magnitudes in nature could be represented through arithmos, the intrinsic properties of whole numbers. This means that for any two magnitudes, one should always be able to find a fundamental unit that fits some whole number of times into each of them (i.e., a unit whose magnitude is a whole number factor of each of the original magnitudes)—an idea known as commensurabilty. Hipassus argued against this idea by demonstrating that for some magnitudes this simply isn't the case—they are incommensurable. Although his original argument is lost to the ages, the following proof, which uses algebraic notions that would have been unfamiliar to the Greeks, gives a sense of the discovery that changed Greek mathematics forever.
Let's imagine a square with a side of length a and diagonal of length b.
If these lengths are commensurable, as Pythagoras and his followers believed (without proof), then there is a common unit u such that a = mu and b = nu for some whole numbers m and n. We can assume that m and n are not both even (for if they were, it would indicate that the common unit could instead be u, and we would simply make that adjustment). So, we can safely assume that at least one of these numbers is odd.
Applying Pythagoras' theorem to the triangle formed in the square, we have:
a^{2} + a^{2} = b^{2}
That is,
2a^{2} = b^{2}
or, substituting our common unit expressions for the two lengths,
2m^{2}u^{2} = n^{2}u^{2}
We know that our common unit, u can't be zero, so we can cancel the u^{2} term from both sides of the equation, leaving:
2m^{2} = n^{2}
Obviously, n^{2} is even, because it is equal to some number, m^{2}, multiplied by two. If n^{2} is even, then n must be even also (if n were an odd number, then n^{2} would be odd). We can express the even number n as two times some number.
n = 2w
Substituting this expression for n into the preceding equation gives us:
(2w)^{2} = 2m^{2}
4w^{2} = 2m^{2}
m^{2} = 2w^{2}
This reveals that m^{2} is a multiple of two, that is, an even number. Consequently, as we reasoned before, m must also be even, and we can write:
m = 2h
Now we have found a contradiction! Remember, we assumed at the beginning
that either m or n was odd, yet we have just shown that both have to be even. This logical contradiction proves that there is no common unit, u, that fits a whole number of times into both a and b—therefore, a and b, the lengths of the side and diagonal of a square, are incommensurable.
• Incommensurable quantities are not rationally related, because this logically leads to an infinite regress.
What does incommensurability have to do with infinity? A contemporary of Hipassus, Theodorus of Cyrene, proved the incommensurability of the side and diagonal of a square by showing that no matter how small of a unit one uses to measure the side and the diagonal, it will never fit a whole number of times into both. In fact, selecting smaller and smaller units merely leads to an infinite regression of triangles. Theodorus' approach is illuminating in that it is more in line with how the Greeks thought about mathematics than the previous demonstration of incommensurability.
To get a sense of Theodorus' proof, let's again focus on the isosceles right (also commonly called a "45°-45°-90°") triangle formed by two sides and the connecting diagonal of a square. Designating this as triangle ABC, with legs of length a and hypotenuse length b, let's once more assume that there is a fundamental unit of measurement capable of representing the lengths of both a side and the hypotenuse in whole number multiples; that is:
a = mu and b = nu
Along the hypotenuse of the right triangle, we can measure a length equal to the side's length and construct a new 45°-45°-90° triangle CDE as shown:
Without too much difficulty, we can show that all three segments, BE, ED, and DC are congruent. (We won't go through the proof, but you would begin by constructing a line from A to E and showing that the two triangles ABE and ADE are congruent.) and that each of these lengths is b-a, again a whole number of copies of u.
Thus, from any 45°-45°-90° triangle with sides whose measure is a multiple of u, we can construct a smaller 45°-45°-90° triangle with sides whose measure is also a multiple of u. We can keep doing this for a number of iterations.
Eventually, however, we will obtain a 45°-45°-90° triangle so small that the length of each of its sides is less than u, which can't be—u was supposed to be the fundamental unit! We might be tempted to think that perhaps u was too big to be the fundamental unit. Using a smaller unit, however, would only delay the inevitable fact that at some point we will reach a triangle with sides whose lengths are shorter than our fundamental unit. Choosing ever smaller units leads to ever smaller "terminal" triangles for as along as we care to continue the process, another example of potential infinity. Our beginning assumption that there was a common unit of measure leads to an absurdity.
We have seen two different ways of demonstrating that the diagonal of a square is incommensurable with its side length. It is not uncommon today to calculate that if the side length of the square is 1 unit long, then its diagonal is units long. The Greeks, themselves, may not have agreed that something such as this is a number. Recall that the Pythagoreans viewed numbers as discrete collections of atom-like units. This view of numbers requires that we have a whole number "counter" to determine the size of the collection and a whole number "namer" to sit in the denominator of the ratio and designate the size of the unit. However, poses a problem because it is not useful in this method; it does not allow us to use whole numbers to serve as "counters" and "namers." This concept put the Pythagoreans in a bind, because it demonstrates that the length of the diagonal of a unit square cannot be a number; consequently, "all is not number." If we insist that such a number must exist because it measures a magnitude that actually does exist, then it is clear that we do not know what a number really is. We shall return to this problem a bit later in the text. The incommensurability argument essentially shows that there are no whole numbers m and n such that =. We call quantities like these, "irrational," and we have seen that their existence is fundamentally linked to a manifestation of infinity (the infinite regress of Theodorus' proof, for instance.) In the previous section, we saw that any rational number can be written as a repeating decimal and vice versa. However, it doesn't take much thought to conceive of a decimal that does not repeat any finite digit sequence and does not end, such as:
0.101001000100001…
Putting aside for a moment the question of whether or not something like this actually exists, we can say at least that this thing cannot be rational, because if it was it would repeat itself, which it is clearly not going to do. Its decimal expansion extends to infinity with no repetitive elements. This brings us to the point that any non-repeating decimal is non-rational, or irrational. It can also be shown that, like the , any square root of a number that is not a square number, will also be irrational. Values such as , , , etc., are all irrational.
Shortly after Hipassus made his arguments for incommensurability, which would lead to the discovery of irrational quantities, an Eleatic philosopher, Zeno, would also show the absurdity of a world in which there were fundamental smallest units of space and time. Recall that the Eleatics held beliefs somewhat diametrically opposed to those of the Pythagoreans—that multiplicity, the idea that the universe is composed of fundamental parts—is ridiculous. They believed in continuous magnitudes in which any perceived boundaries were illusions. This idea is somewhat similar to the concept that "all is one." Similarly to Hipassus' argument for incommensurable magnitudes, Zeno would show that treating a line as a multitude of individual points was philosophically contradictory. These ideas would force thinkers to confront notions of actual infinity—an infinity contained in a limited space—which would prove to be both a powerful concept and a troublesome idea in mathematics.
Next: 3.4 Zeno's Paradoxes